Apparatus for illustrating and demonstrating the principles of pure arithmetic



F. A. RODDY.

y mNclPLEs of Puna ARITHMET APPLICATION FILED APH. 20, 1918.

Patented Mar. 22, 1921.

3 SHEETS-SHEET l.

4APPARATUS FOR ILLUSTRATING AND DEMONSTRATING THE P 1,372,087.

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F. A. Roon/Y. APPARATUS FOR ILLUSTRATING AND DEMONSTRATING THEPRINCIPLES 'OPPURE uw.. .H9 E1 m2,` T ma u M n m P.

APPLICATION FILED APH..20, 191B. 1,372,087.

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F. A. RODDY. APPARATUS FOR ILLUSTRATING AND DEMONSTRATING THE PRINCIPLES0F PURE ARITHMETIC.

APPLICATION FILED APR. 20, T918. 1,372,087'.

3 SHEETS-SHEET 3.

www

Patenged Mar. 22, 1921.

UNITED STATES PATENT orFrcE.

FRANCES AYRES RODDY, F NEW YORK, N. Y.

APPARATUS roniLLUsrnATING AND nnrroivsfraarms THE `rnrncrrnns or PUREanrrnivrn'rrc.

Specification of Letters Patent. Patented Mal', 22, 1921.

Application filed April 20, 1918. Serial No. 229,775.

ff o all w from t may concern y Re it known that I, FRANCES Arens loner,ot New Yoik,Ne\v York, have in `vented an Apparatus for lllustrating andDemonstrating the Principles of Pure Arithmetic, ot which the followingdescription, in connection with the accompanying drawings, is aspecification, like characters on the drawings designating like parts. Yi

My invention relates to a number table7 that is to say, an apparatus bywhich various properties of numbers may be illustrate/d and taught in avery simple, original, and vivid` manner. More particularly, my numbertable is an apparatus for illustrating and demonstrating the principleswhich underlie the composition and divisibility ol numbers, or theprinciples o' pure arithmetic. l

A satisfactory and preferred form of my apparatus is shown in theaccompanying drawings, in Which- Figure 1 is a front view ot theapparatus showing the structure of the double-duov decimal table madevisible to the eye in the arrangement o' the colored unit balls D, alsothe process ot forming numbers outot their prime constants, or primefactors as they are commonly called. The divisors of every number aredenoted by the small figures under the number and Within the saine unitsquare E,

Fig. l is a diagramma-tic view ol the arrangement of complementarydivisors on an enlarged scale which could not be slioivn clearly in Fig.l.

Fig. 2 is a vieu7 ot the reverse side of the apparatus, showing thearrangement ot the 'tour classes olI numbers in the double-duodeeimaltable and the formulae Gad-1, (5a-l- Gn-l-EB, (Sn-kil, Gal-5.1 (la,under which they are contained.

The small figure under each number denotes the sum oli' its digits.

il.. hole a in which one of the kindergarten pegs is inserted whenrequired, is seen above each number and Within the same unit square.

Fig. 2a shovvsa kindergarten peg, lettered 19.

Fig. 3 `s a view of the adjustable slate which is adapted to be usedwith the apparatus `shown in Figs. l and 2. `This slate consists otparallel horizontal bars B'. Each bar has a silicate surface and isruled in `lnventy-four squares E', the saine as the bars of thedouble-duodecimal table. i

This slate is used for teaching and illustrating the principles oit purearithmetic as set forth in the annexed specilications. When it isrequired, it is suspended over the doubleduodeeimal table and fastenedby pegs 77 inserted through registering holes a in the respectivecorners of the slate and table.

According to the preferred embodiment of my invention, the apparatuscomprises a iframe A provided with alternately arranged bars B and slotsC. Each bar B is ruled in @el squares E on which are placed appropriatenumerals or otherindiciu to be hereinafter referred to. The squares lllon the bars of the table and the squares l ou the bars ot the slate areoit the same are: Every slot C contains strung on the rods lltwenty-:tour balls oreounters D arranged te register With thel squares Eet the slate. which, as shown in Fig. 3, is composed of the frame il',the bars B and tl e slots The sideI ot the unit square and the diameterolf the unit ball or counter are the same and serve as alinear unit inmeasuring. cemparing, combining and separating numbers.

How I use the cloublc-thtotlec1im(il tabl@ a cache/ng beg/zomers.

In my double-duodeimal. table 'the 'lfo'rnis olf all numbers ivitliregard to Ql (the base ol' the table) and Witlrregard to every exactdivisor of 24 are vividly illustrated by means ot colored balls l).rlhere are twenty four. in each rovv sliding on a horizontal rod F, Theyare arranged to register with tbe series of squares E in which areinscribed the figures which show. the number of units each numbercontains. `llVhen the balls in each row are moved to the left, the firstball is opposite l, the second opposite 2, the third opposite 3, thefourth opposite 4, and so on.

The side of the unit square and the diameter of the unit ball thus serveas linear units in measuring and comparing numbers, fractional parts,lines, &c.

PERFECT NUBIBER PICTURES.

The balls D in each row represent the six standard colors; red, blue,yellow, purple, orange and green. They are arranged as follows: two red,two blue, two yellow, two purple, two orange, andl `two green; and whenthey are all moved to the left as far as possible, we have a vividpicture of odd and even numbers, with the exact number of 2s in everyeven number, and the exact number of' 2s plus l in every odd numberillustrated yin contrasting colors. Thus,

. The number 2 is pictured by two red balls.

The 2 and l in the number 3 are pictured by the two red balls and theone blue ball.

The two 2s in 4 are pictured by two red balls and two blue balls.

' The two 2s and l in 5 are pictured by vtwo red, two blue balls and oneyellow ball.

' The three 2s in 6 are pictured by two red, two blue and two yellowballs.

The three 2s and l in 7 are pictured by two red, two blue, two yellow,and one purple ball; and so on.

Since this arrangement of these colored balls shows the exact number of2s in every even number and the exact number of 2s plus l in every oddnumber, it is obvious that it must afford the perfect picture of everynumber with all its integral parts, whether addends or divisors, vividlypictured in contrasting colors.

Odd numbers stand in the first column and every second column after thefirst.

ven numbers stand in the second column and every' second column afterthe second.

The stmcture of the clouble-aluodecz'mal 1fable and how vt s illustratedby the me 'rangement of the colored balls.

In the double-duodecimal table there are twenty-four numbers in eachrow; hence, there are twenty-four vertical columns, and

Vall numbers that stand in the same column are of the same form withregard to 24; that is, when divided by 24, they will all yield the sameremainder..

Every number greater than 24 is either exactly divisible by 24 or whenthe division isvcarried as far as possible there will be a remainder ofl, 2, 3, 4, 5, 6, 7, 8, 9, 10, ll,

All numbers exactly divisible by 24 stand in the twenty-fourth column.When a nurnber is not exactly divisible by 24 the remainder will be thefirst number or" the column in which it stands. If the number stands inthe first column the remainder will be l; if it stands in the secondcolumn the remainder will be 2; if it stands in the third column theremainder will be 3; and so on.

All remainders are pictured in the arrangement of the colored balls.

When the remainder is l we have one red ball, when it is 2 we have twored balls, when it is 3 we have two red balls and one blue ball, when itis 4 we have two red halls and two blue balls; and so on.

THE FORMS or ALL NUMBnns wrrii incolla) 'ro EVERY nxiicr nivIson or 24.

The divisors of 24 are l, 2, 3, 4, 6, S, l2 and 24.

Since all numbers of the same form with regard to 24 stand in the samecolumn, all numbers of the same form with regard to any divisor of 24must stand in the same column or columns.

If the first number of any column is a multiple of 2, every number is amultiple of 2.

If a multiple of 3, every number is a multiple of 3.

Similarly, if thc first number of any column is a multiple of 4, G, S orl2, every number in the column is a multiplo respectively of 4, G, 8 or12.

The multiples of 2 are the numbers in the 2d, 4th, 6th, 8th, 10th, 12th,14th, 16th, 18th, 20th, 22nd and 24th vertical columns.

The multiples of 3 are the numbers in the 3rd, 6th, 9th, 12th, 15th,18th, 21st and 24th columns, and similarly the multiples oi' 4, (j, 8and l2 are the numbers in those columns which are respectively the 4th,6th, 3th aud 12th columns.

When the unit balls are all moved to the left as far as possible, wehave a vivid pic ture of the arrangement of the columns con` taining themultiples of each divisor of 24.

When a number is a multiple of 2, the two right hand balls are eitherred, blue. yellow. purple, orange or green.

Then a number is a multiple of 4- or 3, the two right hand balls areblue, purple or green.

lhen a number is a multiple of (l. the two right hand balls are yellowor `green.

Then a number is a multiple of l2 or 24. the two right hand balls aregreen.

lVhen a multiple of 3 is an odd number. theright hand ball is eitherblue or orange: and when it is an even number, the two right hand ballsare either yellow or `green.

When the first number of any column is not divisible by a given divisorof 24. tlu` remainder will be the same as the remainder which everynumber of that column will yield when divided by that divisor.

' For example. 3 is l more than 2; therestand.

`fore, every number in the third column divided by 2 Will leaveremainder l; 5 is 2 more than 3; therefore, every number inthe iithcolumn divided by 3 will leeve u. remainder of 2; 7 is: 8 more then el;therefore, every number in the seventh column divided by fl will leavesi. remainder oi 3; 7 is l more than 6; therefore, every number in theseventh Column divided by 6 will leave e remainder 1.

When the first number in :my column is less than e given diviser ei 2li,it will be the remainder which every other number iu the column willyield when divided b y that divisor.

For exemple: 1 is less then 2; therefore, every number after l in theirsteelumn divided by 2 Will yield e remainder l; 2 is lese than 3;"therefore, every number :liter 2 in the seeond column divided by 3 willyield it remainder 2; Sie less than Li; therefore. every number eiter 3in the third eelumn divided by 4 vvill yield n remainder lVhen anynumber is not divisible by there will be :i remainder l; when it is notdivisible by 3, there ivill be :i remainder l er 2; when it is notdivisible by 4, there will be e remainder l, 2 or E; when itis notdivisible by G. there will be e. remainder 1, 2, 3, el, er 5; when it isnot divisible by 8, there will be e remainder i, 2, 3, Ll, 5, (i er 7 gwhen it is not divisible by 12, there will be urenniinderi,fz,e,i,5,e,7,e,9,10m-ii.

The infrange/:itent of lle columns contr/,tiring Hl@ odd and the @eenmultiples of every @effet dfi/visor of 2.4,.

An odd multiple ei2 number is ene that eontziins it sin edd number oftimes und un even multiple is one that Contains it :in even number oittimes.

2 G 10 are odd multiples et 2.

l 8 l2 ere even multiples et 2. l2 36 60 are odd multiples of: l2. 2-fl4:3 72 nre even multiplesbi l2.

The number 2l cont-eins 2, Si, fil-. und l2 un even number et times;therefore, the edd und even multiples of these divisere eredietinguished by` the columns in which they lf the first. number of anyeelumn :in odd multipleeii 2, all the numbers nre edd nuiltiples oi' 2;it the iirst number is un even multiple oi? 2e :ill the numbers nre evenmultiples oi' 2; ii' the tiret number is :in edd multiple et 3, all thenumbers :ire odd multipms oi 3 il" the iirst number ie :in even mnltpleof 8, ell the numbers ure even multiples oil" 3; and so on.

lil/'lien the unit bells in erich row ure moved to the lei'it :is ter nepossible, the arrangement oi the edd und. even multiples of 2, 3, el, Gend l2 is vividly pictured in the contres-ting colors.

l l Red Bue Yellow Pulrple Oreiige Grnn When i number is en edd multipleof 2, the tivo right hand bells nre ene of the lighter colore, eitherrei, yellow or orange; end `when it en even multiple of 2, or e multipleoi 4;, the tive right hand bells ere one ol' the darker colors, eitherblue, purple or green. i

When u number is en odd multiple o'lv (i, the two right hund ballsl nreyellonT und when it is an even multiple et' G or e multiple oi l2, 'thetive right bend bulls nre lgreen.

lVhen n. number is en odd multiple of 3,

the righthzuid bull is either' blue er erunge,

und. when it nu even multiple oi2 3, or :i multiple of (l, the tiverigrht hund bulls nre yellow or green.

When e number is u multiple ol l2, the tivo right hund bulls :ire green.

Principle *is 'not f main l) ei flu odd imdtfz'ple of (my mim/ Zier ZJ/eZrj/ un mien @Multiple of Hint The urinni'ement et the eoluninecentaininp; the edd :ind even multiples oi 2 und Ll shevv that un oddmultiple et 2 ezinnot be divided by 4l, und :in edd multiple ol 4iezinnet be divided by S.

The nrrungement of the Columns eentnining; the odd und even multiples ei3, G and i2, shoeT that :in odd multiple oi 3 eennot be divided by il,en odd multiple ei 6 cannot be divided by 1.2,end en `edd multiple et l2cannet be divided by i How the Mimes of the niwnbens in, @Ueli eolimmt(rre menmrie'erd und 241 are 25 and 4:8 :ireA il!) und 72 :ire 7?;

und 9G ere 97 und 120 :ire 121 and 144: are 145 Y AN EXPLANATION OF THEALGEBRAIC FORBIULIE ON THE .MARGIN OF TIIE DOUBLE-DUODECIMAL TABLE ANDHOV THEY ARE USED IN TEACH- ING THE PRINCIPLES XVHICH NDERLIE THEOOIWIPOSITIONAND DIVISIBILITY OE NUBIBERS, OR THE PRINCIPLES OF PUREARITHBIETIO.

In the series of integral numbers we have an organized system ofcontinual proportionals.

The system is of the nature of a system of measures. Four unit measuresof proportion form its fundamental basis; hence, there are four distinctclasses of numbers and foury denominations.

. The four unit measures are l, 2, 3, and 6. 6 is the standard unit. l,2, and 3 are the smaller units.

VThe unit measure of the first class is l.

The unit measure of the second class is 2.

The unit measure of the third class is 3.

The unit measure of the Jfourth class is G.

The ls (numbers of the rst class) are the powers of 5, 7, ll and thehigher prime numbers, and the numbers in the successive geometricalprogressions which are formed by multiplying ltogether these powers. *i

The 2s (numbers or" the second class) are the powers of 2, and thenumbers in the successive geometrical progressions which are formed bymultiplying together the powers of 2, and the powers ot 5, 7, lll andthe higher prime numbers.

The 3s (numbers or" the third class) arethe powers of 3 and the numbersin the suecessive geometrical progressions which are formed bymultiplying together the powers yof 3, and the powers of 5, 7, ll andthe higher prime numbers.

The Gs (numbers of the fourth class) are the numbers in the successivegeometrical v progression which are formed by multiplying together thepowers of 2, the powers of 3 and the powers of the higher prime numbers.

VThe numbers in each class are distinguished by their form with regardto 6.

l/Vhut I vmea/.n by the form of a. number with regard to 6.

Every number greater than 6 is either eX- actly divisible by 6, or whenthe division is carried on as far as possible. there will be a remainderof l, 2, 3, 4t, or 5; that is, every number whatever is of one of theforms of 6u, 6n+l,'6n-|2, (Sufi-3, (fm-t4, Gn-l-.

lhen the remainder is the number is Y contained under the formula(Sn-l-l.

When the remainder is 2, the number is containedv under the formula(Sn-t2, and so on, z'. e. whatever the remainder may be, the number iscontained under the formula n-l-remainder unless there is no remainder,

in which case the number will be contained under the formula (in.

The system is embodied in the formula GnGn-l-l, 6n+2, 6ft-t3, Grt-l-t,Git-ITF).

ihe complete development oi` the i'ormulae and the distinct propertiesof cach class require that the series of integral numbers (l, 2, 3 )bearranged in six vertical columns instead of one straight line. 7Whenthis is done, we have what may be called the senary table.

In the senary table all numbers of the same form with reg'ard to G standin the same column and the key to the system is found in the dispositionand arrangement of the si): columns and in multiplying, dividing.addingr and subtractingr the formulxe.

'IIIE SENARY TABLE.

1st class. v M

- ith class. Col. C01. ol Col. Col. Col. I II Hl IV VI 67i+1 G11-t2 (M+S071+: 6;:1-5 611 2 3 1 5 o S 9 1u 11 12 14 15 l lo 1T 1S i 20 2l y 22 232l l 2o 27 l 2s zo :io 32 It! 35 I 3G as as io n i:

T/m arrangement of zf/zc fom' classes 0f mwubcrs.

l number of this class is 2 more or 2 less than a multiple of G, and is.therefore, contained under the formula (ln-t2 or (hifi-4. Thosecontained under the formula (Sn-t2 stand in the second column; thosecontained under the :Formula Gel-,L4 stand in the fourth column.

The third class contains all odd numbers that are measured by 3. Everynumber ot this class is more or 3 less than a multiple of 6. and is,therefore, contained under the formula (Sn-l-S. All numbers of thisclass stand in the third column.

The fourth class contains all even numbers that are measured by 3. Thenumbers` of this class are contained under the. vformula (iny and standin the sixth column.

The arrangement of the four classes vol numbers in the double duodecimaltable is as follows:

In the double duodecimal table the series 1, 1, 1% is arranged intwenty-four vertical columns. lfwenty-four 1s an even multiple of 6;therefore, in the structure of this mulas under which they are containedon the margin of the table. See Fig. 2.

The divisibility of a. number by 6 is indicated by the final sum of itsdigits. `Vhen the digits of any number are added, the final sum obtainedwill be either 2, 3, fl, 5, 6, 7, 8, 9 or 16. Then a number is 1 more or4 more than a multiple of 6, the final sum of the digits will be eitherIl, 7 or 10. Vhen the number is 2 more or 5 more than amultiple of 6,the final sum ofthe digits will be either 2, 5 or 8. Then a number is amultiple of 6 or 3 more than a multiple of 6, the final sum of thedigits is either 3, 6, 0r 9. Hence, possible and impossible powers areindicated by the final sum of their digits. Thus, when the final sum ofthe digits is 4.-, T, 9 or'10,the number is a possible power of anydegree. Then the final sum of the digits is 2, 5 or 8, the number is apossible odd power of any degree but cannot be a square. When the linalsum of the digits is 3 or 6, the number cannot have an integral root.This property is seen in the multiplication of the formulae. i

How to` 'use the double Zuorlccz'mal table a #cac/eng the process offorming 7mm-bers out of their prima constants or' prima factors as theyare commonly called.

In the drmvingit will be observedthat there are small figures beneatheach number within the same unit square and these small figuresillustrate the process of forming numbers out of their prime factors.The process of measuring and recording the mul tiples of prime numbersinvolves the fol-` Vof the two or more prime numbers,`

The process of forming numbers out of their prime constants is thegeneral process ef pure. arithmetic, and when it `1s carriedsufficiently far and fully analyzed, it will be seen to contain all thespecial processes which have their origin 1n the use of the primenumbers as constant multipliers, such as factoring, commonmultiples,common divisors, involution, evolution, progression, etc.

This process is my discovery and` it is, therefore, necessary to exglainthe succes sive steps in the process before I Aan show how they aretaught` and illustrated.`

The multiples of prime number are distinguished by their forms withregard to the powers of the prime number, that is, by the number oftimes the prime constant is used in their formation. Thus, when theprime constant is used once, the number is a multiple of the primenumber only; when it used twice the number is a multiple of the square;when it is used three times the number a multiple of the cube; and soon.

In the process of forming,` the multiples of a prime number, 1 is thefirst multiplicand; therefor, I write 1 under every number and withinthe same unit square. Then I measure the multiples of the successivepowers of the successive prime numbers in the order in which they areformed from unity.

The successive steps in this process are as follows: i

The multiples of 2.-T he irst prime number is 2. Takingr 2 and everyysecond number after 2, I have the multiples of 2: 2, 4, 6, 8, 10, etc.Y

The multiples of the powers of 2 Aare arranged as follows: y p Everysecond multiple of 2 is a multiple of 41 (the square) every secondmultiple of 1: is a multiple of 8 (the cube) every second multiple of Sis a multiple of 16 (the fourth power) 5 every second multiple of 16a'multiple of 32 (the fifth power); and

so on.

Therefore, I write The multiples of 3.--The first number after 2 that isnot measured by 2 is 3; therefore, flis the second prime number.` Takingil and every third number after 3, I have themultiples of 3: 3, 6, 9,12, 15, 1S, `21, 27, etc. i

i In measuring,V the multiples `of the powers .of all other primenumbers `which enter into the table, i. c., 5, 7, 11 `and the higherprime numbers` I proceed in a similar manner.

When this is done, I have under every number and within `the same unitsquare, the

one or more prime constants of the number `and the one orimore powers ofeach prime f .95, 9s, 99, 10o, at.

consta-nt, according to the number of times that prime constant is usedin forming the number. Y

When a number has but one prime constant, I place the one or more powersof the prime constant in a horizontal line to the rightot 1. Thus: 1, 2,4.

When a. number has two or more prime constants, l place the one or morepowers ot the first or smallest prime constant in ay horizontal line tothe right of 1, and the one or more powers-of the one or more largerprime constants in a verticalline under 1, in the order of theirmagnitude:

Thus: 1, 2, 4.

` lWhen'a number has two prime constants, T multiply the one or morepowers of the first or smaller prime constant in succession by the oneor more powers of the second prime constant in succession, and place thesuccessive products formed by each `multiplier in a horizontal line sothey coincide prime Constantin succession, and place the successiveproducts formed by each multiplier in parallel horizontal lines so thatthey coincide with the'multiplicands.

lNhen a number has tive or more prime constants i proceed in a similar.manner.

The numbers that have two or more prime constants are:

6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 36, 3S, 89,40, 42, 44, 45, 46, 5o, 5i, 52, 54., 55, 5c, 57, 5s, ce, ce, es, 65, 66,68, 69, 70, 72, 74, 75, 76, 77, 78, s2, sa, 85, se, 87, se, 9o, 91, 92,93, 94,

I begin with 6 and take these numbers in succession, multiplyingtogether the one or morepowers of the two or more prime constants undereach number before passing to the next. Then this is done the divisorsof vevery number stand under the number and within the same unit square.They are arranged inthe order in which they are formed out of theirprime constants. All divisors that are formed by using the smallestprime constant a given number of times` stand in the same column, andall that are formed by using any larger prime constant a given number ottimes stand in the same row or rows, and hence` the number oit timeseach prime constant is used in terming any divisor is indicated by thecolumn and the row in which it stands.

The greatest common divisor ot two or more numbers is the number formedby using each of their common prime constants the least number of timesthat it is used in any one of the numbers; therefore, it is .indicatedby the column and the row in which it stands.

The least common multiple ot two or more numbers is the number formed byusing each of their common prime constants the greatest number ot timesit is used in any one of the numbers; therefore, it is indicated by thecolumn and the row in which it stands.

The square of a number is the. number formed by using each oi'l itsprime constants twice the number of times it is used in the number; thecube is the number Yt'ormed by using each of its prime constants threetimes the number of times it is used in the number; the fourth power isthe number formed by using each of its prime constants tour times thenumber oit times it is used in the number; and so on; therefore, everypower of the number is indicated by the row and the column in which itstands.

The square root of a number is the number formed by using each ot itsprime constants half as many times as it is used in the number; the cuberoot is the number formed by using each of its prime constants one thirdthe number of times it is used in the number; and so on. Therefore` theintegral root of a perfect power is indicated by the row and the columnin which it stands.

From the foregoing explanations, it will be seen that the process offorming the divisors ot a number, the common divisor ot two or morenumbers, the least eonnuon multiple oli two or more numbers.` the powersot a number, the integral roiot ot a perfect power. Src., are allcontained in the general process` of formingnumbers out ot their prime-onstants or prime factors.

The divisors of every number are so written that the distance betweenthe sueeessive divisors in each horizontal row is the same as thedistance between the sueeessive divisors in each vertical column;therefore.

the divisors of every number 'torni either a horizontal line, a squareor an oblong, according to the number of its prime constants and thenumber of times each prime constant is used.

their the number is a power o l a prime number, its divisors 'torni ahorizontal line.

lllhen a niunber is a common multiple of only two prime numbers and eachprime constant is used the same number ot times, there are as manyhorizontal rows as there are vertical columns and these form a square,while the divisors oli' every `other nui'nber :torni an oblong. lVhenthere are more rows than eolinnns, the oblong is horizontally disposedand when there are more columns than rows, the oblong is verticallydisposed. lrlence the different kinds ot numbers can be distinguished atsight.

rrI-in AnnANonMnN'r or ooirrLnMEN'rAnY 1n visons.

'll he divisors of a number are arranged in pairs so proportioned thatthe product oi? each pair equals the number.`

These pairs are determinedby the laws of position, or by the laws otsimple and compound geometrical progression, which are as AliolloirslWhen there is but one series oit divisors,

the product o'll the lirst and last divisor is equal to the product otany two divisors equally distant from them vand to the square of themiddle divisor when there is an odd number of divisors.

When there are two or more series ot divisors, the product of thelirstdivisorof the first series and the last divisor ot the last series isequal to the product ot any two divisorsequally distant 'from them andto the square ot the middle divisor when there is en odd number ofdivisors.

lf call these pairs complementary divisors oil? the number.

lnde'volopingpthe lawsot position, l. and the number itself must` beregarded as the first pair oi complementary divisors. Every other pairstand either horizontally, vertically or diagonally opposite and equallydistant :trom the first pair.

lVhen a number is asquare it has an odd number ol divisors; and hence,al midi'lle divisor which multiplied by itseli equals the number. Thismiddle divisor is thew square root. lilyery divisor, except the squareroot, has its complementa-ry divisor, which stands horizontally,vertically or diagonally opposite and equally distant from the squareroot. Y lllhen a, nunber `contains two series et divisors, every divisoro'tthe first series has its complementary divisor in 'the second.lil'lhen a number contains three se ol di- \iio s. every divisor of thel'irst series has its oomplementary divisor in the third, and

every divisor in the second series has fits `romplementary divisor inthe second. Vhen the number contains :tour series of divisors, -yerydivisor ot the first series has its oom- J:Ylementary divisor` in thetourth, every -diriser o ...the second `series has its complementarydivisor in the third; and so on.

Any two numbers kening fon their product L ef/nce number areconziplementory` di risers of that number; hence, when the nir/m- 71er scli/vided 0?/ @it/wr (Zai/visor the other will be Quotient.

rl'lie arrangement oit' the complementary 70 divisors 'oli a number isshown by means olf lines connecting each pair in the 'followingexamples.

Complementary divisors Division. of 1G: 1o ,zie

l 8 lo l I Complementary divisors 1 z i s 5 10 2o fio The complementaryparts of a number may be shown by means oli linesconnecting1 eaeh pair,as in the following illustrations: r Complementary parts ot 3.

Complementary parts ot 4.

Complementary parts of 5.

r2 3 ii 5 l-low l use the adj ustablo slate in teachingv andillustratiiig the analysis oil. a number.

l` write l il il in the lirst row, l 3 a; in the seeond, l 2 3 5 in thethird, and so on.` l. ronnect the eomplementary parts or `eiufhsi'lrcessive number with lines and write a ud illusl conneet THE SUBI`O1"` IWVO NUMBERS.

The" sum of two numbers is the same in whichever order they are added.

Illustratie@ er1/emplea F or brevity three dots are here used :for theword therefore. Thus,` l-l-Qzf. Q-l-lz 1s read l plus 2 equals,therefore, l. 2 plus 1 equal 3. i

Complementary parts oit 3.

:i y1+i:3,-.2i1=3 Complementary parts o'tet. 1 2 a i lrfwl I show by theuse of the balls that the sum ot any two numbers 1s the same in 130Whichever order they are added, and Write each pair on the adjustableslate; thus,

Complementary parts of 3.

Complementary parts of 4.

THE DIFFERENCE BETXVEEN TWO NUBIBERS.

Any two `numbers Whose sum is equal to a given number are complementaryparts ot that number. It the smaller number is l, they are the iirstpair; if 2, they are the second pair; if 3, they are the third pair; andso on.

When one of the complementary parts is subtracted from the number, theother will be the remainder.

Illustrated examples.

Thus, 1-l-3:4.`.41:3 is read one plus three e uals four; therefore, tourminus l equals t ree. f

Complementary parts of 3.

Complementary parts of 4.

i 2 3 `4 1+3=4..4-3= 1anu41=3 2+2=4..42=2 Complementary parts of 5.

l 2 3 4 5 1 u l I show by the use of the balls that when one of thecomplementary parts is subtracted from the number, the other Will be theremainder, and Write each pair on the adjustable slate; thus,

Complementary parts of 3.

Complementary parts of 5.

PRINCIPLES.

1. The sum oftivo even numbers is even.

2. The sum of two odd numbers is even.

3. The sum of an odd number and an even number is odd.

-Hence, if one of the complementary parts of an even number is even, theother must be even; if one is odd, the other must be odd.

.If one of the com lementary parts of an Y odd number is odd, he othermust be even.

Y Illustrative examples.

An analysis of the numbers 6 and 7 amply serves to illustrate thecomplementary parts of even and odd numbers respectively and how in theuse of my table these complementary parts are clearly comprehended andvlsualized.

Complementary parts of 6.

l 2 3 4 5 G l '-1 l Complementary parts oiz 7.

1 and 5 are odd. 2 and 4 are even.

l is odd. e is even. 2 is even. 5 1s edd. 3 1s odd. y11s even.

Complementary parts of 7.

II ou) I use t/ee double clumlectfmal table u 'teueltug themultiplication (zml (tiefsten tables.

When the numbers 1, 2, 3, etc., are each multiplied by the same number,the successive products are called ay multiplication table.

When the multiplier is 2, the successive products are the multiples et 2and the table is called the two times table; when the multiplier is 3,the successive products are the multiples oit 3 and the table is calledthe three times tableg and so on.

The divisors of 24- are 2, 3, 4, G, S, 12, 24. Taking these divisors inthe order in which they are formed out ot the prime constants 2 and 3,We have 2, 4, S, a geometrical series of three numbers, and 3, 6, l2,24. a geometrical series of four numbers.

Every second multiple of 2 is a multiple of 4; every second multiple of4 is a multiple of S.

Every second multiple of 3 is a multiple of 6; every second multiple otG is a multiple of 12; every second multiple ot l2 is a multiple of 24.

These seven multiplication tables are taught in the order in which theyare formed out of their prime constants 2 and 3. First, the two times.four times and eight times tables; then, the three times. six times,twelve times and twenty-four times tables.

The successive products in each table are measured by the unit balls andWritten on the adjustable slate.

The two times table-I move the unit balls, tivo at a time, to the left,saying 2 times l are 2, 2 times 2 are 4, 2 times 3 are 6, and so on. IWrite the iirst product (2) under the second ball, the second prod lll()uct (4) underthe fourth ball, the third product (G) under the sixthball; and soon.

2 is contained in 24 twelve times; therefore,` the multiples ot 2 standin twelve columns, viz., 2d, 4th, .6th, 8th, 10th, 12th, 14th, 16th,18th, 20th, 22d and 24th.

The fou/r' times miler-I move the unit balls four at a time to the lett,saying 4 times 1 are 4, 4 times 2 are 8, 4 times 3 are 12, and so on. Iwrite the iirst product (4) under the fourth ball, the second product(8) under the eighth ball, the third product (l2) under the twelfthball; and so on.

4 is contained in 24 siX times; therefore, the multiples of 4 stand insix Columns, viz., 4th, 8th, `12th, 16th, 20th and 24th.

I proceed in a similar manner with respect to the eight times table, thethree times table, the six times table and the twelve times table,moving the unit balls as many at a time as are indicated by the tableemployed.

When the unit balls in each row are all moved to the left as `tar aspossible, the arrangement o' the columns containing the products in eachof these multiplication tables is vividly pictured in contrasting colorsand the products in each of these tables stand in parallel verticalcolumns.

When a number Vis an odd multiple ot 2, the two right hand balls areeither red, yellow or orange,` and when it is an even multiple of 2, thetwo right hand balls are either blue, purple or green.

The odd multiples of 2 are the numbers in the 2d, 6th, 10th, 14th, 18thand 22d columns. The evenfmultiplesof 2 are the numbers in the 4th, 8th,12th, 16th, 20th and 24th columns. Ihe even multiples of 2 are themultiples of 4.

Theodd multiples of 4 are the numbers in the 4th, 12th' and 20thcolumns. 'Ihe even multiples of 4 are the numbers in the 8th, 16th and24th columns. The even mul tiples of 4 are the multiples of '8.

The multiples of 8` are `the numbers in the 8th, 16th and 24th columns.In 24 (the base of the double-duodenary table) 8 is contained an oddnumber oftimes; therefore, the `odd and even multiples of 8 are notdistinguished by the columns in which they stand but are all containedin the 8th, 16thand`24th columns.

The odd multiples of 3 are the numbers in the 3d, 9th, 13th and 21stcolumns. The even multiples of 3 are the numbers in the 6th, 12th, 18thand 24th columns. The even multiples of 3 are the multiples of 6. When anumber` is an odd multiple of l6, thetwo right hand balls are yellow,and

when 1t 1s an even multiple of 6, the two right hand balls are green.

The odd multiples of 6 are the numbers in the 6th and 18th columns. Theeven multiples of `6 are the numbers in the 12th and 24th columns. Theeven multiples oit G are the multiples of 12.

The multiples of 12 `are the numbers in the 12th and 24th columns. Theodd multiples -o't' 12 are the numbers in the 12th column. The evenmultiples of 12 are the numbers in the 24th column. The even multiplesof 12 are the multiples 0111.24. When a number is a multiple of 12 or24, the two right hand balls are green.

rIhe following diagrams show how the unit balls are arranged inmeasuring the successive products in each of these multiplicationtables, and how each successive addend is pictured in contrastingcolors. The colors are `designated by RR (red), BB `(blue), YY (yellow),PP (purple), 00 (orange), GG (green).

The 2 times table.

The 8 times table.

OOGGRRBB 00000000 XYPPOOGG 00000000 00000000 V00000000 00000000 0000000000000000 The 3 times table.

The 6 Limes table.

. PIOOGG RRBBYY 000000 000000` 000000` 000000 000000 000000 000000000000 The IZ times table.

RRBBYYPIOOGG RRBBYYIICOGG 000000000000 000000000000 000000000000000000000000 000000000000 000000000000 Proces oooooo oooooo ooooooooooco oooooo The successive products in every multiplication table froman arithmetical progression; hence, the sum of the first and thirdproducts isequal to twice the second or middle product; the sum of thefirst and fourth products is equal to the sum of the second and thirdproducts; the sum of the first and fifth products is equal to the sum ofthe second and fourth products and equal to double the third or middleproduct; and soon. I

v I show the arrangement of these pairs by connecting the extremes andythe means with lines. I write the sum and the difference of each pairon the adjustable slate, and show their relative magnitude by the use ofthe unit balls, as in the following illustration.

The two times tabla-The successive products in the two times table forman arithmetical progression in which the common difference is 2, viz.:2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24. I move the unit balls, twoat a time, to the left, repeating the products and the commondifference; thus, 2 and 2 are 4, 4 and 2 are 6, 6 and 2 are 8, 8 and 2are 10, 10 and 2 are 12, 12 and 2 are 14, 14 and 2 are 16, 16 and 2 are18, 18 and. 2 are 20, 20 and 2 are 22, 22 and 2 are 24.

T he f combinationa 10-2= 8 and 10- 8=2 -10-4= 6 and 10- 6=4 12-2=10 and12-10=2 12-4= 8 and 12- 8=4 12-6= 6 HOW KINDERGARTEN PEGS ARE USED INFIND- ING PRIME NUMBERS.

. a in each unit square, and in these holes l insert Lkindergarten pegswhenever desired in illustrating the principles of arithmetic.

In finding prime numbers I insert a peg p in the hole under the primenumber 2 and write 2 under every second number after 2.

When this is done, the first number after 2 that is not marked 2 is 3;therefore 3 is a prime number.

I insert a peg in the hole under the prime number 3, and write 3 underevery third number after 3. When this is done, the rst number lafter 3that is not marked 2 or 3 is 5; therefore 5 is a prime number.

I vinsert a peg in the hole under the prime number 5, and write 5 Linderevery fifth number after 5. When this is done, the first number after 5that is not marked 2, 3 or 5 is; therefore 7 is a prime number.

I insert a peg in the hole under the prime number 7, and write 7 underevery seventh number after 7. Vhen this is done, the first number afterT that is not marked 2, 3, 5 or 7 is 11; therefore 11 is a prime number.

I proceed in a similar manner in showing the arrangement of the largerprime numbers and their multiples.

Before beginning this process, I move all the balls to the left as faras possible. The peg under every prime number is the same color as theright hand ball of the prime number. Thus, I place a red peg under 2 andunder 13; a yellow peg under 5 and under 17 a purple peg under 7 andunder lf);

a green peg under 11 and under 23; and so on. t

Fractional parte.

Fractional parts are illustrated with the unit balls and written on theblank squares of the adjustable table.

rlhus, move up six balls and every ball represents one-sixth of G.Divide the balls into groups of 2s and every group represents one-thirdof 6. Divide the balls into groups of 3s and every group representsone-half of 6.

On the iirst row show the sixths.

On the second row show the halves.

On the third row show the thirds. Thus,

Whoa o o o Halves (I) O IO 'rimas qho'CI) p` Picture all the fractionalparts at the same time so they can be measured and compared.

What part of 6 is 1? What part of 6 is 2? What part of 6 is 3? What partof 6 is 4? What part of 6 is 5? lVrite the fractional parts in the blanksquares.

of G. of G.

dial

1. of 2. of 3. of

Show by the use of the balls that twothirds 1s equal to four-sixths;thus (I) (I) O O O TWO-thirdS.

o o o (l) o o Four-sinus.

I Show by representing the fraction with lines that two-thirds is equal*to fourthsixths.

llO

wise than as set forth in the claims read in connection with thisspecification.

Vhat I claim as new and desire to secure by Letters Patent is 1. Anumber table provided with balls of standard colors arranged to show theforms of all numbers with regard to the radix 24 and with regard to eachdivisor of said radix.

2. A number table provided with balls of standard colors arranged toshow the forms oi' allnumbers with regard to the radix 24 and withregard to each divisor of said radix, the arrangement of these ballsforming a vivid picture of all the parts of every number whether addendsor divisors which can be seen without moving the balls.

3. A number table provided with balls of standard colors arranged toshow the forms of all numbers with regard to the radix 24 and withregard to each divisor of said radix, the arrangement of these balls-forming a vivid picture oi' all the parts of every number whetheraddends or divisors which can be seen without moving the balls, saidnumber table being provided with a surface ruled in squares.

4. A number table provided with balls of standard colors arranged toshow the forms oi' all numbers with regard to the radix 24 and withregard to each divisor of said radix, the arrangement of theseballsforming a vivid picture of all the parts 0i every number whether addendsor divisors which can be seen without moving the balls, said numbertable being provided with a surface ruled in squares, the side of the`square and the diameter of the ball serving as a linear unit inmeasuring, comparing, combining and separating numbers.

5. A number table provided with balls of standard colors arranged toshow the forms of all numbers with regard to the radix 24 and withregard to each divisor of said radix, the arrangement of these ballsforming a vivid picture of all the parts of every number whether addendsor divisors which can be seen without moving the balls, said numbertable being provided with aL surface ruled in squares, the side of theSquare and the diameter of the ball serving as a linear unit inmeasuring, comparing, coIn bining and separating numbers, said surfacebeing provided with holes for receiving pegs of standard colors.

6. Apparatus ior demonstrating the pring ciples of arithmetic7 saidapparatus comprising a frame with a series of supporting members eachhaving a set of movable counters, and frame members having indicia ofthe numerical relationships appertaining to said counters inpredetermined relative positions, said indicia including sockets andvari-colored pegs for insertion in said sockets to accentuate saidnumerical relationships.

Signed at New York in the county of New York and State of New York, thiseighteenth day of April, 1918.

FRANCES AYRES RODDY. Witnesses:

ALEX. C. PRoUDFrr, Gao. H. CONGER.

